Problem 3. Let A be a complex nxn (n ≥2) normal matrix. Suppose that λ = 1 is an eigenvalueof A satisfy |μ< 1. Proveof A of (algebraic) multiplicity one and that all other eigenvaluesthat the sequence of powers of A converges entrywise to a nonzero matrix of the form xx*, i.e.,lim Ak = xx* for some nonzero à E Cn.k-400

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Answered: Problem 3. Let A be a complex nxn (n…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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4. Let A be a square matrix. (a) 1² is an eigenvalue of A? with the same eigenvector, v. Show that if A is an eigenvalue of A with corresponding eigenvector v, then (b) What is the characteristic polynomial of A2? Suppose that the characteristic polynomial of A is fa(1) = (A + 2)(A – 3). Suppose that v is an eigenvector of A². Is it necessarily true that v is also (c) an eigenvector of A? If so, then prove this fact; if not, then give an example that shows this is false.
VII. Run two iterations using normalized power iteration method for the following matrix and initial vector. Roughly predict the dominate eigenvalue and corresponding eigenvector. A = [ ²₁ 2] ₁ x₁ = 1 []
1.5.3 (a) If the eigenvalues of the n x n matrix A are λ > λ > 13... λ = 0 show that the eigenvalue λ, can be found by applying the power method to the matrix kl - A, where is the identity matrix and k > 2₁.
in 2) Suppose that A is a square matrix and let p(x) be a polynomial with real coefficients with variable x. If p(x) = E a;x' i = 1 in , then define p (A) = E a¡A' . If, is an eigenvalue of A, the show that p (1) is an eigenvalue of p(A). i = 1
6. Consider the matrix A = Compute the eigenvalues of A, and the corresponding eigenspaces.
A = To 1 0 101 010 Consider an Nx N matrix A with N orthonormal eigenvectors x' such that Ax' = x¹, where the A, is the eigenvalue corresponding to eigenvector x'. It can be shown that such a matrix A has an expansion of the form: A =Σ/x)(x| = Σ\x(x)". i) Show that if the eigenvalues are real then A, as defined through the above expansion, is Hermitian. ii) Using the result for A show that the Nx N identity matrix can be written as 1=Σx'(x¹)¹. i=1
4. Let At be a matrix that depends on r and is given by At t 3 -2 1 (a) For which te R matrix At has no real eigenvalues for At? (b) For which t E R matrix At has exactly one real distinct eigenvalue? Give that eigenvalue. (c) For which te R matrix At has two different real eigenvalues? Give one such eigenvalue as a function of t.
4. A real, symmetric 3-by-3 matrix M has eigenvalues X₁ = 4, X₂ 1 vector corresponding to the eigenvalue X₁ = 4 is v₁ -1 0 [4] -2 to the eigenvalue X₂ = 6 is v₂ = 6 and X3 = -3. An eigen- and an eigenvector corresponding What is M?
4. The eigenvalues of a matrix A € Cnxn are usually not computed in practice by finding the zeros of the characteristic polynomial. One way to approximate the eigenvector corre- sponding to the eigenvalue of a diagonalizable matrix A with largest absolute value is power iteration, Axi ||Axi||2 Xi+1 i = 0, 1, 2, ..., where o C is some initial guess. Approximation for the eigenvalue of A with largest absolute value is computed as Hi = R(A, x₂), i = 0, 1, 2, ... (1) The intuition behind power iteration is that repeated multiplication by A turns the vector slowly towards the eigenvector corresponding to the eigenvalue of A with largest absolute value (unless xo is orthogonal to it). Let now [2 1 0] A = 1 2 1 0 1 2 Choose o [1,0,0]¹. Compute an approximation for the eigenvalue of A with largest absolute value Xamax (A) and the corresponding eigenvector by using power iteration. Plot the error Xamax (A) — μ| as a function of the index i = 0, 1,..., 10.
7. Let the 2 x 2 matrix A have real, distinct eigenvalues and μ. Suppose that an eigenvector of A is (1,0)" and an eigenvector of u is (-1,1)". Sketch the phase portraits of is x = Ax for the following cases: (1) 0 0. وچستان ستان ج
The matrix. has an eigenvalue of multiplicity 2 with corresponding eigenvector U. Find A and U. A = λ = has an eigenvector v = -13 -5
22. Let P be an idempotent matrix. Show that the only eigenvalues of P are 1 Suppose that Px = Ax, x + 0.] 0 and 1 = 1. [Hint:

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